Infeger Programming Production I in~ Problems

No. 847 April 1969

Infeger Programming Production in~ Problems

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ND ELDON

L, NORMAN

Ir ..."d"'''''I1/ fll /,'ul'l'sfl'y and Conservat'ion

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PlUNCIPAL FUNCTIONS ofa is the preparation of cutting 'I'11l':4u sehedules, essential to efficient III 111 ~. transform the allowable cut deci­ I \ 'II 'j,inj.!," plan of action, Therefore, the 111111 WeI' usually attaches significant im­ 1111' 'II Ilwi I' development, If the manager 11 I' Ii, :~('lll'c111ling problems in terms of III 1IIIi;c.inj.!," or minimizing an objective • II Id,jl'ct to a set of constraints, it is pos­ III 1'1111 to dl~velop optimum schedules over .t 1.1 Illning' period. This, however, requires II III \' "illl mathematical programming pro­ IH

I 11\;llIaJ.!:'CI'

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hilI'

llescribes a potentially useful programming technique known as /ll 'I 11111':\1' programming. While no new inte­ I pI (r"llllllni'llg (IP) algorithm is developed, a I III JpJl "I' several known IP algorithms is un­ I I ,1·1"1, IllId one of the more recent ones is sub­ 1'1111 II, 1\ i1)lplied to forestry oriented problems. , 1'11"11)' Ii (,olllputational problems have ham­ I ., I d I hi' prndi. lem prohIbIt more than one compartment from IWjlll~' har,,:ested in. any given cutting period. TherefoJ'(', ('0))·' stra1llt equatlOns (2) and (3) become redundant anI'! LIII' problem can be solved by using a transportation aig-oriLhm. However, if different compartment acreages arc chosen it is likely that it will not be possible to formulate th is as a transportation problem. Thus! for increased flexibility,

sample problem Oill! was l).ot consldered as II t1'1lnSpol'tHtion problem.

4

2

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n

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'001 volume per compt. Ithousand cu. ft.)

1l1'l'iMing to obtain the final Ill' h a short time since both 1"l'el;rnan (1965) report that II " Ii ro r i lhm seems to work well II If I 'illgllp to 30 variables. In ad­ II ,It I 111 WHS fairly well constrained, I • I IH,V\' ..;;eems to increase compuII

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the number of compartments cutting periods to ten. This III J It ('hf'duling problem involved 100 , 11 I II'. ,:llld 30 constraints. After 30 'I 1(' ()r)()() central processor time no "lilli, n \vas obtained. Even more alarm­ tIll! r ';\sible solution (obtained in 38 ) \ II \ inlin 9% of the value obtained Ill. 11 Iddilional 1762 seconds of computing. III IIhltll'lll, t he last solution obtained was only llhlll ,., r M t.he known optimal feasible solu­ " 111111 l' ,d'

t I III.

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11111, shown in Table 2, it was observed (lljl.:dive function was quite insensitive II t!ll!l'l' '1:1 harvesting schedules. Thus, a sub­ I 1\ lit! I'" I'd-ring of the harvest schedule only It ,Ill. iiII'· ·ted the value of the objective funcI ,11. 'I'hi" riiay indicate that an intuitive subjec­ 'h 'uul(~ is not too far from the optimum. 'I lli' 11 "1 lid sample scheduling problem concerns II II 1'1 sL \·vhic.:h has been subdivided into ill stands ,'tfh I'll('h stand being subjected to n possible har~ "M i, v: alteTnatives. Let ru equal the residual llhll' of stand i if harvesting alternative j is ,o.;t·I,·!'tl'd. and let Xu equal 1 if stand i is subjected

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Table 2, Optimal schedule for sample problem one, Cutting Period

Compartment to Harvest

1 2 3 4

4 2 1

5

5

3

to harvesting aJtcmative j, and 0 otherwise. Henr.e", we wish to :

Table 3. Solution comparisons for sample problem two

(7) In addition, there is a maximum volume which may be removed during the harvesting operation. Let Vij equal the volume available for harvesting from stand i if harvesting alternative j is se­ lected. To maintain an adequate growing stock, a harvest volume of no more than Y bd.ft. may be removed. This gives rise to to the following inequality:

(8) There is an additional constraint concerning the stumpage value of the stands to be harvested. Let Sjj equal the stumpage value of stand i if harvested by alternative j. The stumpage val ue for the entire harvesting operation must be great­ er than or equal to Z dollars. Writing this an an inequality we obtain: III

. ~ 1

1=

. ~ 1

J

=

Sij

(9)

Two additional constraints are needed to complete the problem formulation. Each stand must be com­ pletely harvested if any cutting is done in the stand, and Xji must equal either 1 or 0, i.e., equa­ tion set (4). Assume that we have 10 stands with 9 harvest­ ing alternatives per stand. This gives rise to a problem involving 90 binary variables and 12 con­ straint equations. For reasons noted in equations (5) and (6), we must increase the number of con­ straints to 22. Further, we have a maximum har­ vesting volume of 850,000 bd.ft. and we must sell a minimum of $10,000 worth of stumpage. For this problem we know the optimum LP solution, but not the optimum IP solution. Using the same computer program version of Geoffrion's algorithm referred to above, an opti­ mum solution was not found after 20 minutes of computing. However, the incumbent solution after 12.5 seconds was only 3.667r less than the opti­ mum linear programming solution. It is not known whether this is the optimum integer solution since all solutions were not implicitly enumerated in the 20-minute time limit set for the run. Be­ cause the optimum integer solution must be less than or equal to the optimum LP solution, this IP solution may be optimal. Table 3, which contains the LP and IP harvest­ ing schedule.s for this problem, illustrates several interesting aspects of the problem. In comparing

Stand

Optimum LP Schedule Alternative Number

4 3

1

2

3

I

4 5

6 3-51 ('t, 4-49";,

6 7 8 9 10 Max

4 4

3 2 3 ,. ~i

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r ij

Xu

Residual Value

Geoffrion Schedule Alternative Number

Residual Value

5,625 2,899 4,798 4,324 599

6 5

6,950 1,321 4,66 ] 4,324 0

3,155 8,533 4,177 2,239 2,398

6 3 3 7 3

2,485 9,737 4,177 1,415 2,398

$38,983

1 7

3

$37,468

-­

the LP and Geoffrion iOolutions, it again appearR that the objective function iR not particularly sen­ sitive to the variableI' in the solution set since the Geoffrion schedule produced a similar value of the objective function with only three of the ten selected variables in common. This indicates that a large number of near optimum solution setR probably exist. Conclusions

To completely evaluate the performance of some of the proposed integer programming algorithms it would be necessary to solve many different types of scheduling problems. Moreover, the two spe­ cific harvest scheduling problems discussed in this bulletin should be solved by some of the other integer programming algorithms. However, even this approach would not necessarily permit one to draw general conclusions concerning the efficiency or usefulness of a specific algorithm for solving the harvest scheduling problem. The reasons for this are: (a) one algorithm may be more efficient for solving one specific formulation while a second algorithm may be more efficient for solving an­ other formulation, and (b) the efficiency of the proposed algorithms is directly related to the size of the problem. However, as Freeman (1965) states, "as with any enumeration procedure com­ putational experience is the best indicator of its worthiness." Thus, additional computational ex­ perience may be warranted for the purpose of determining the most efficient algorithm to use

for a specific formulation of the hllfvllflt f!cheduI­ ing problem.

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The impjioal II II' 110 "I es1;" inte­ ger progl':Ir1IIIl!11 'I I' Jill '\(On I'llI' problems of a similar xil'IIl'IIII't, "Ill. II' Ii.· fulilre will reveal a method :-lin1 III' I I It hllpl. pl'o('cc!ure which will ha\'1' \ ILI(' lippi,' IlJlil,. Ilnwpvcl', the trend seel1l:-t 10 IJ\.: [n iiI ',IIII) il1lpli ·il ('numeration tech­ niquPI-\ \ hidl od (J'Y ,jJil'ielltly for problems or it '1'1'1:1!11 Mlllll'[III'I', hut very inefficiently for problf'lIJH , " II dill' 'I'('ut structure. In addition, c01ll1 1I :11.i0l1jtl 11m '1-\ V:ll'y almost exponentially wit.h lhl' /'liz.' oj" f 11(' pl'obJem.

Twp 1'11l'tJllragi Ilg" items which we observed (:1) all initial feasible integer solution was (1)1:lill'd 'III a VHy short time, and (b) the value of thl' 01 .i ,dive function associated with the ini­ tial ~-i(>IIl! ion was often quite close to the value of U1I' (,lJ.i 'dive function associated with the opti­ l1WIl1 :.;olution. However, the problem containing 100 bi nary variables did not follow this pattern bC('(llll'; ~ the initial and final (not optimal) objec­ I.i ve function values were not even close to the optimum value. A major limitation of the implicit enumeration approach is due to the fact that the optimum is unknown, even though it may have been found, before all solutions have been implicitly enumer­ ated. Although it is difficult to generalize on the basis of two tests, this suggests that one couijd merge an implicit enumeration algorithm with another IP solution procedure to obtain a more efficient computational algorithm. This observa­ tion was also noticed by Freeman (1965). \VOl":

Another observation is that integer program­ ming tends to place more emphasis on careful problem formuTation. Hence, to reduce the num­ ber of decision variables, attempts should be made to delete alternatives which are of minor imp 01'­

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tance to the problem. Also, as previously noted, the Geoffrion algorithm increases the size of the problem when equality constraints are used. In addition, and more important computationally, is the fact that Geoffrion's algorithm is designed fur minimization problems. Therefore, when maxim­ ization problems of the type discussed in this bul­ letin are encountered, a transformation (Xi = 1 - Xi) is required. This increases the number of variables in a partial solution, and thus slows down the computational efficiency of the algo­ rithm, Methods for alleviating this problem are currently under study. It is apparent that, at present, the relatively large nature of practical forest scheduling prob­ lems and the efficiency of existing integer pro­ gramming algorithms are not compatible. How­ ever, the characteristics of the general structure of many forest scheduling problems indicate that the problem is amenable to solution. Since the coefficient matrix is tied over all variables by relatively few constraints, and since the matrix tends to be uniform and independent in blocks across the variables, it lends itself to decomposi­ tion. The potential of thi::-; procedure should be evaluated. From the preceding description of the constraint matrix, it follows that the density is generally low. Simplex algorithms have taken advantage of this characteristic, but as yet implicit enumera­ tion methods have not. We concur with Lemke­ Speilberg (1967) who feel that the area of im­ plicit enumeration is in the infant stage and eventually will become much more successful as more sophisticated search methods are developed. In addition, the newer filter methods and the probability approach should be evaluated through computational experience.

Literature Cited Balas,E. 1965. An Additive Algorithm for Solving Linear Programs with Zero-One Variables. J. Operations Res. 13(4) :517-549. Balas, E. 1967. Discrete Programming by the Filter Method. J. Operations Res. 15 (5) :915-957. Balinski, M. 1965. Integer Programming: Methods, Uses, Computation. Mgt. Sci. 12 (3) :253-313. Benders, J. F. 1962. Partioning Procedures for Solving Mixed-Variables Programming Problems. Numerische Mathematik 4 :238-252. Curtis, F. 1962. Linear Programming the Management of a Forest Property. J. For. 60:611-616. Dantzig', G. B., D. R. Fulke son, and S. Johnson. 1~Jfj4. Solution of a Large-Scale Traveling-Salesman Prob­ lem. J. Operations Res. 2: 393-410. Dantzig, G. B. 1959. Note 011 Solving Linear Programs in Integers Naval Res. Logistics Ql'tly 6 :75-76. Donnelly, R. H., R. W. Gardner, and H. R. Hamilton, 1963. Integrating Woodlands Activities by Mathematical Programming. Battelle Memorial Institute (for Amer­ ican Pulpwood Association). Freeman, R. 1965. Computational Expel'ience with the Balas Integer Programming Algorithm. RAND Corp. 1'-3241. Geoffrion, A. 1967. Integer Programming by Implicit Enu­ meration and BalaH' Method. SIAM Review. 9 (2) :178­ 190. Geoffrion, A. 1968. An Improved Implicit Enumeration Approach for Integer Programming. Western Mgt. Sci. Inst. Wol'1